Commutative Algebra

Job: local–global algebra of commutative rings — Spec, localization, Noetherian geometry, dimension, and depth — as the engine of algebraic geometry and arithmetic.

Core

Commutative algebra is the internal language of schemes: primes are points, localization is “zooming in,” and Noetherian / depth / dimension control finiteness and singularities.

Layer Content
Spec / localization primes, \(S^{-1}R\), local rings, Nakayama
Noetherian package ACC, Hilbert basis, primary decomposition
Dimension Krull dim, catenary, principal ideal / Krull theorems
Depth / regularity regular sequences, Auslander–Buchsbaum, Serre criteria
Homological pd, gl.dim, Koszul, local duality pointer

Reading map

Page Focus
Foundations Spec intuition, localization, Noetherian, primary decomp, dim, depth, Serre, completion, henselian
Homological commutative algebra projective/global dimension, Auslander–Buchsbaum, Koszul, local duality pointer

Dependency order

  1. Rings and modules — rings, modules.
  2. Foundations (this track) through dimension and regular sequences.
  3. Homological chapter \(\leftrightarrow\) homological algebra.
  4. Apply to schemes and AG foundations.

Interfaces

Area Contact
Algebraic geometry \(\operatorname{Spec}\), stalks, smooth/regular, proper via valuative
Homological algebra Ext/Tor, pd, dualizing complexes
Algebraic number theory Dedekind domains, DVRs, completions
Representation theory invariant theory, Hilbert–Samuel

Notation (paper-standard)

  • \(\operatorname{Spec} R\), \(R_\mathfrak{p}\), \(\mathfrak{m}\)-adic completion \(\hat R\)
  • \(\dim R\), \(\operatorname{depth} M\), \(\operatorname{pd}_R M\), \(\operatorname{gl.dim} R\)
  • \(\operatorname{Ass}(M)\), primary decomposition; regular / Cohen–Macaulay / Gorenstein

See it

Smoothness algebraically. A finitely presented \(k\)-algebra \(A\) is smooth of relative dimension \(d\) iff for all primes the local rings after geometric base change are regular of the expected dimension — equivalent to the Kähler differentials \(\Omega_{A/k}\) being projective of rank \(d\) plus a lifting criterion (infinitesimal). Serre’s \((R_k)+(S_k)\) package and the Jacobian criterion live entirely in commutative algebra before one writes “smooth morphism of schemes.”

Check

Why does Noetherian alone not imply unique factorization of ideals?

Noetherian domains need not be Dedekind (dim may be \(>1\), not integrally closed). Primary decomposition replaces unique prime factorization of ideals; only in Dedekind domains do nonzero ideals factor uniquely into primes.

In one sentence: what does \(\operatorname{depth} M < \dim M\) detect?

Failure of the Cohen–Macaulay condition — “too few” regular sequences relative to dimension, typically from embedded components or singular structure (e.g. coordinate rings of non-CM varieties).

Pitfalls

  • Maximal ideals vs all primes: Nullstellensatz identifies max ideals over alg. closed fields with points; arithmetic and generic points need all primes.
  • Localization kills Tor/Ext information only after base change carefully; flatness is the correct “no higher Tor” condition.
  • Completion is faithful for f.g. modules over Noetherian local rings (Krull) but can introduce pathology if non-Noetherian.

Refs

  • Matsumura, Commutative Ring Theory
  • Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry
  • Atiyah–Macdonald, Introduction to Commutative Algebra
  • Stacks Project, Commutative Algebra
  • Bruns–Herzog, Cohen–Macaulay Rings

Next

Foundations · Homological CA · Modules · Rings and fields · AG foundations · Homological algebra